The innovative geometry of European Space Agency Swarm constellation opens the way for new investigations based on magnetic data. Since the knowledge of a vector field on two spherical surfaces allows calculating its curl, we propose a new technique to estimate the curl of the ionospheric magnetic field measured by Swarm satellites A and B, orbiting the Earth at two different altitudes from March to September 2014. Using this technique, we mapped the amplitude of the radial, meridional, and zonal components and of total intensity of the ionospheric current density at the satellite's altitudes, i.e., the F region of the ionosphere, during two local nighttime intervals: before and after midnight. Most of the obtained results are consistent with some of the known features of nighttime F region currents; others need further investigation. The proposed technique could contribute in selecting magnetic data with minimum contamination from nighttime F region electric currents for magnetic modeling purposes.